Triple Angle Formulas/Cosine/Proof 3

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Theorem

$\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$

Proof

$\ds \cos 3 \theta$

$=$

$\ds \cos \paren {2 \theta + \theta}$

$\ds $

$=$

$\ds \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta$

Cosine of Sum

$\ds $

$=$

$\ds \paren {2 \cos^2 \theta - 1} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta$

Double Angle Formula for Cosine and Double Angle Formula for Sine

$\ds $

$=$

$\ds 2 \cos^3 \theta - \cos \theta - 2 \paren {1 - \cos^2 \theta} \cos \theta$

Sum of Squares of Sine and Cosine after algebra

$\ds $

$=$

$\ds 2 \cos^3 \theta - \cos \theta - 2 \cos \theta + 2 \cos^3 \theta$

multiplying everything out

$\ds $

$=$

$\ds 4 \cos^3 \theta - 3 \cos \theta$

gathering terms

$\blacksquare$

Sources

1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid